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Search: MSC category 14G35 ( Modular and Shimura varieties [See also 11F41, 11F46, 11G18] )

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1. CJM Online first

Koskivirta, Jean-Stefan
Congruence Relations for Shimura Varieties Associated with $GU(n-1,1)$
We prove the congruence relation for the mod-$p$ reduction of Shimura varieties associated to a unitary similitude group $GU(n-1,1)$ over $\mathbb{Q}$, when $p$ is inert and $n$ odd. The case when $n$ is even was obtained by T. Wedhorn and O. B?ltel, as a special case of a result of B. Moonen, when the $\mu$-ordinary locus of the $p$-isogeny space is dense. This condition fails in our case. We show that every supersingular irreducible component of the special fiber of $p\textrm{-}\mathscr{I}sog$ is annihilated by a degree one polynomial in the Frobenius element $F$, which implies the congruence relation.

Keywords:Shimura varieties, congruence relation
Categories:11G18, 14G35, 14K10

2. CJM 2013 (vol 66 pp. 924)

Stankewicz, James
Twists of Shimura Curves
Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livné on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

Keywords:Shimura curves, complex multiplication, modular curves, elliptic curves
Categories:11G18, 14G35, 11G15, 11G10

3. CJM 2013 (vol 65 pp. 1125)

Vandenbergen, Nicolas
On the Global Structure of Special Cycles on Unitary Shimura Varieties
In this paper, we study the reduced loci of special cycles on local models of the Shimura variety for $\operatorname{GU}(1,n-1)$. Those special cycles are defined by Kudla and Rapoport. We explicitly compute the irreducible components of the reduced locus of a single special cycle, as well as of an arbitrary intersection of special cycles, and their intersection behaviour in terms of Bruhat-Tits theory. Furthermore, as an application of our results, we prove the connectedness of arbitrary intersections of special cycles, as conjectured by Kudla and Rapoport.

Keywords:Shimura varieties, local models, special cycles

4. CJM 2013 (vol 66 pp. 1167)

Rotger, Victor; de Vera-Piquero, Carlos
Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves
The purpose of this note is introducing a method for proving the existence of no rational points on a coarse moduli space $X$ of abelian varieties over a given number field $K$, in cases where the moduli problem is not fine and points in $X(K)$ may not be represented by an abelian variety (with additional structure) admitting a model over the field $K$. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired on the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided $\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$. We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.

Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstruction
Categories:11G18, 14G35, 14G05

5. CJM 2011 (vol 64 pp. 1248)

Gärtner, Jérôme
Darmon's Points and Quaternionic Shimura Varieties
In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.

Keywords:elliptic curves, Stark-Heegner points, quaternionic Shimura varieties
Categories:11G05, 14G35, 11F67, 11G40

6. CJM 2010 (vol 62 pp. 668)

Vollaard, Inken
The Supersingular Locus of the Shimura Variety for GU(1,s)
In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

Categories:14G35, 11G18, 14K10

7. CJM 2009 (vol 61 pp. 828)

Howard, Benjamin
Twisted Gross--Zagier Theorems
The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate heights of complex multiplication points on the modular curve $X_0(N)$ (and on Shimura curve analogues) with the central derivatives of automorphic $L$-function. We extend these results to include certain CM points on modular curves of the form $X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article "Central derivatives of $L$-functions in Hida families", Math.\ Ann.\ \textbf{399}(2007), 803--818.

Categories:11G18, 14G35

8. CJM 2008 (vol 60 pp. 734)

Baba, Srinath; Granath, H\aa kan
Genus 2 Curves with Quaternionic Multiplication
We explicitly construct the canonical rational models of Shimura curves, both analytically in terms of modular forms and algebraically in terms of coefficients of genus 2 curves, in the cases of quaternion algebras of discriminant 6 and 10. This emulates the classical construction in the elliptic curve case. We also give families of genus 2 QM curves, whose Jacobians are the corresponding abelian surfaces on the Shimura curve, and with coefficients that are modular forms of weight 12. We apply these results to show that our $j$-functions are supported exactly at those primes where the genus 2 curve does not admit potentially good reduction, and construct fields where this potentially good reduction is attained. Finally, using $j$, we construct the fields of moduli and definition for some moduli problems associated to the Atkin--Lehner group actions.

Keywords:Shimura curve, canonical model, quaternionic multiplication, modular form, field of moduli
Categories:11G18, 14G35

9. CJM 2002 (vol 54 pp. 352)

Haines, Thomas J.
On Connected Components of Shimura Varieties
We study the cohomology of connected components of Shimura varieties $S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\olomega$ on the group of connected components of $S_{K^p}$ we define an operator $L(\omega)$ on the cohomology groups with compact supports $H^i_c (S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual trace of the composition of $L(\omega)$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi^r_p$, can be expressed as a sum of $\omega$-{\em weighted} (twisted) orbital integrals (where $\omega$-{\em weighted} means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\olomega$). As the crucial step, we define and study a new invariant $\alpha_1 (\gamma_0; \gamma, \delta)$ which is a refinement of the invariant $\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

Categories:14G35, 11F70

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